$X$, $Y$ i.i.d r.v's. Prove that $\mathbb{E}[X\mathbb{1}_{\{X+Y \in B\}}] = \mathbb{E}[Y\mathbb{1}_{\{X+Y \in B\}}] $

Question

Let $X, Y$ be i.i.d random variables with finite expected values. I want to justify that $$ \int_{\{x+y \in B\}}x\mu(dx)\mu(du)=\int_{\{x+y \in B\}}y\mu(dx)\mu(du). $$

I would appreciate any hints, tips or explanation.

Answer 1

Define the function $f\colon\mathbb R^2\to\mathbb R$ by $$ f(x,y)=x\mathbf 1_B(x+y). $$ The equality you want to show is $\mathbb E\left[f(X,Y)\right]=\mathbb E\left[f(Y,X)\right]$. To this aim, use the following facts:

1. The random vector $(X,Y)$ has the same distribution as $(Y,X)$.
2. Hence $f(X,Y)$ has the same distribution as $f(Y,X)$.